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A cubic Dirac operator and the emergence of Euler number multiplets of representations for equal rank subgroups. (English) Zbl 0952.17005

This is the detailed description of results in “The Weyl character formula, the half-spin representations, and equal rank subgroups” [B. Gross, B. Kostant, P. Ramond, and S. Sternberg, Proc. Natl. Acad. Sci. USA 95, 8441-8442 (1998; Zbl 0918.17002)]. Let \(\mathfrak{p}\) be a finite-dimensional complex vector space, \(\mathcal{B}_{\mathfrak{p}}\) be a nonsingular symmetric bilinear form on \(\mathfrak{p}\), \(\mathfrak{r}\) be a complex finite-dimensional Lie algebra, and \(\mathcal{B}_{\mathfrak{r}}\) be a nonsingular \(\text{ad }\mathfrak{r}\)-invariant symmetric bilinear form on \(\mathfrak{r}\).
Let \(\nu: \mathfrak{r}\to \text{Lie }SO(\mathfrak{p})\) be a \(\mathcal{B}_{\mathfrak{p}}\)-invariant representation of \(\mathfrak{r}\) on \(\mathfrak{p}\). Let \(\mathfrak{g} = \mathfrak{r}+\mathfrak{p}\). At first, a representation \((\nu, \mathcal{B}_{\mathfrak{p}})\) of Lie type is defined and a criterion for a Clifford algebra to be of Lie type is given. Next, a cubic Dirac operator is defined and studied for a Lie type representation \((\nu, \mathcal{B}_{\mathfrak{p}})\). Hereafter \(\text{rank }(\mathfrak{r}) = \text{rank }(\mathfrak{g})\) is assumed. \(\dim(\mathfrak{p})\) is even, so that the Clifford algebra \(C(\mathfrak{p})\) is simple and the spin module \(S\) is irreducible. For the definition of “multiplet” and its roles, see p. 479. Infinitesimal character values are given on p. 488.
Finally, let \(G\) be a connected and simply connected compact Lie group with \(\text{Lie }(G) = \mathfrak{g}_{\mathbb{R}}\) and \(R\) be the subgroup of \(G\) with \(\text{Lie }(R) = \mathfrak{r}_{\mathbb{R}}\). Note that \(\dim X = \dim G/R\) is even. An ”equal weight assumption” for \(\mathfrak{g}\) and \(\mathfrak{r}\) is introduced and it is shown that it is satisfied if and only if \(X = G/R\) is a spin manifold (i.e., the second Stiefel-Whitney class of X vanishes).

MSC:

17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
22E46 Semisimple Lie groups and their representations
22E50 Representations of Lie and linear algebraic groups over local fields
58J99 Partial differential equations on manifolds; differential operators

Citations:

Zbl 0918.17002
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References:

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